On the Local Operator Commutation Relations and Extensions of Locally Defined Positive Definite Functions on the Heisenberg Group

It is proved that if a group of unitary operators and a local semigroup of isometries satisfy the Weyl commutation relations then they can be extended to groups of unitary operators which also satisfy the commutation relations. As an application a result about the extension of a class of locally defined positive definite functions on the Heisenberg group is obtained